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Everything is linear to a first approximation
:Everything is linear to a first approximation In , a '''linear approximation' is an approximation of a general using a (more precisely, an ). They are widely used in the method of to produce first order methods for solving or approximating solutions to equations. Definition Given a twice continuously differentiable function f of one variable, for the case n = 1 states that : f(x) = f(a) + f'(a)(x - a) + R_2\ where R_2 is the remainder term. The linear approximation is obtained by dropping the remainder: : f(x) \approx f(a) + f'(a)(x - a) . This is a good approximation for x when it is close enough to a ; since a curve, when closely observed, will begin to resemble a straight line. Therefore, the expression on the right-hand side is just the equation for the to the graph of f at (a,f(a)) . For this reason, this process is also called the tangent line approximation. If f is in the interval between x and a , the approximation will be an overestimate (since the derivative is decreasing in that interval). If f is , the approximation will be an underestimate. Linear approximations for functions of a vector variable are obtained in the same way, with the derivative at a point replaced by the matrix. For example, given a differentiable function f(x, y) with real values, one can approximate f(x, y) for (x, y) close to (a, b) by the formula : f\left(x,y\right)\approx f\left(a,b\right)+\frac{\partial f}{\partial x}\left(a,b\right)\left(x-a\right)+\frac{\partial f}{\partial y}\left(a,b\right)\left(y-b\right). The right-hand side is the equation of the plane tangent to the graph of z=f(x, y) at (a, b). In the more general case of s, one has : f(x) \approx f(a) + Df(a)(x - a) where Df(a) is the of f at a . Applications Optics Gaussian optics is a technique in that describes the behaviour of light rays in optical systems by using the , in which only rays which make small angles with the of the system are considered. In this approximation, trigonometric functions can be expressed as linear functions of the angles. Gaussian optics applies to systems in which all the optical surfaces are either flat or are portions of a . In this case, simple explicit formulae can be given for parameters of an imaging system such as focal distance, magnification and brightness, in terms of the geometrical shapes and material properties of the constituent elements. Period of oscillation The period of swing of a depends on its , the local , and to a small extent on the maximum that the pendulum swings away from vertical, θ0, called the . It is independent of the of the bob. The true period T'' of a simple pendulum, the time taken for a complete cycle of an ideal simple gravity pendulum, can be written in several different forms (see ), one example being the : : T = 2\pi \sqrt{L\over g} \left( 1+ \frac{1}{16}\theta_0^2 + \frac{11}{3072}\theta_0^4 + \cdots \right) where 'L''' is the length of the pendulum and g is the local . However, if one takes the linear approximation (i.e. if the amplitude is limited to small swings, ) the is: : T \approx 2\pi \sqrt\frac{L}{g} \qquad \qquad \qquad \theta_0 \ll 1 \qquad (1)\, In the linear approximation, the period of swing is approximately the same for different size swings: that is, the period is independent of amplitude. This property, called , is the reason pendulums are so useful for timekeeping. Successive swings of the pendulum, even if changing in amplitude, take the same amount of time. Electrical resistivity The electrical resistivity of most materials changes with temperature. If the temperature T'' does not vary too much, a linear approximation is typically used: : \rho(T) = \rho_0(T - T_0) where \alpha is called the ''temperature coefficient of resistivity, T_0 is a fixed reference temperature (usually room temperature), and \rho_0 is the resistivity at temperature T_0 . The parameter \alpha is an empirical parameter fitted from measurement data. Because the linear approximation is only an approximation, \alpha is different for different reference temperatures. For this reason it is usual to specify the temperature that \alpha was measured at with a suffix, such as \alpha_{15} , and the relationship only holds in a range of temperatures around the reference. When the temperature varies over a large temperature range, the linear approximation is inadequate and a more detailed analysis and understanding should be used. References Category:Intermediate mathematics